The rate of change of surface area of a sphere of radius $r$ when the radius is increasing at the rate of 2 cm/sec is proportional to
Answer & explanation
Correct answer: option 4
Let S be the surface area and $r$ be the radius of the sphere at any time t. Then,
$S=4 \pi r^2$
$\Rightarrow \frac{d S}{d t}=8 \pi r \frac{d r}{d t}$
$\Rightarrow \frac{d S}{d t}=8 \pi r \times 2$ [$\frac{d r}{d t}$ = 2 cm/sec]
$\Rightarrow \frac{d S}{d t}=16 \pi r \Rightarrow \frac{d S}{d t} \propto r$